Exceptional points of two-dimensional random walks at multiples of the cover time

Abstract

We study exceptional sets of the local time of the continuous-time simple random walk in scaled-up (by $N$) versions $D_N\subseteq {\mathbb{Z}}^2$ of bounded open domains $D\subseteq {\mathbb{R}}^2$. Upon exit from $D_N$, the walk lands on a "boundary vertex" and then reenters $D_N$ through a random boundary edge in the next step. In the parametrization by the local time at the "boundary vertex" we prove that, at times corresponding to a $\theta$-multiple of the cover time of $D_N$, the sets of suitably defined $\lambda$-thick (i.e., heavily visited) and $\lambda$-thin (i.e., lightly visited) points are, as $N\to \infty$, distributed according to the Liouville Quantum Gravity $Z_{\lambda }^D$ with parameter $\lambda$-times the critical value. For $\theta <1$, also the set of avoided vertices (a.k.a. late points) and the set where the local time is of order unity are distributed according to $Z_{\sqrt{\theta }}^D$. The local structure of the exceptional sets is described as well, and is that of a pinned Discrete Gaussian Free Field for the thick and thin points and that of random-interlacement occupation-time field for the avoided points. The results demonstrate universality of the Gaussian Free Field for these extremal problems.

arXiv comment48 pages, 5 figures

References in corpus (11)

• Extreme local extrema of two-dimensional discrete Gaussian free field
• From loop clusters and random interlacement to the free field
• Full extremal process, cluster law and freezing for the two-dimensional discrete Gaussian Free Field
• Late points for random walks in two dimensions
• Random interlacements and the Gaussian free field
• Two-dimensional random interlacements and late points for random walks
• Planar Brownian motion and Gaussian multiplicative chaos
• Extrema of the two-dimensional Discrete Gaussian Free Field
• On intermediate level sets of two-dimensional discrete Gaussian Free Field
• Thick points of random walk and the Gaussian free field
• Frequently visited sites of the inner boundary of simple random walk range

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• Near-maxima of the two-dimensional Discrete Gaussian Free Field
• A limit law for the most favorite point of simple random walk on a regular tree
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